
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>51</b></div></td><td><div><a href="index.php?section=problems&amp;id=51" style="text-decoration:none;" title="Click to view problem">
Find the smallest prime which, by changing the same part of the number, can form eight different primes.
</a></div></td><td><div style="text-align:center;">5287</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>52</b></div></td><td><div><a href="index.php?section=problems&amp;id=52" style="text-decoration:none;" title="Click to view problem">
Find the smallest positive integer, <i>x</i>, such that 2<i>x</i>, 3<i>x</i>, 4<i>x</i>, 5<i>x</i>, and 6<i>x</i>, contain the same digits in some order.
</a></div></td><td><div style="text-align:center;">14234</div></td></tr>

<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>53</b></div></td><td><div><a href="index.php?section=problems&amp;id=53" style="text-decoration:none;" title="Click to view problem">
How many values of C(<var>n</var>,<var>r</var>), for 1 &#8804; <var>n</var> &#8804; 100, exceed one-million?
</a></div></td><td><div style="text-align:center;">12896</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>54</b></div></td><td><div><a href="index.php?section=problems&amp;id=54" style="text-decoration:none;" title="Click to view problem">
How many hands did player one win in the game of poker?
</a></div></td><td><div style="text-align:center;">6164</div></td></tr>

<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>55</b></div></td><td><div><a href="index.php?section=problems&amp;id=55" style="text-decoration:none;" title="Click to view problem">
How many Lychrel numbers are there below ten-thousand?
</a></div></td><td><div style="text-align:center;">11213</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>56</b></div></td><td><div><a href="index.php?section=problems&amp;id=56" style="text-decoration:none;" title="Click to view problem">
Considering natural numbers of the form, <i>a<sup>b</sup></i>, finding the maximum digital sum.
</a></div></td><td><div style="text-align:center;">12105</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>57</b></div></td><td><div><a href="index.php?section=problems&amp;id=57" style="text-decoration:none;" title="Click to view problem">
Investigate the expansion of the continued fraction for the square root of two.
</a></div></td><td><div style="text-align:center;">7778</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>58</b></div></td><td><div><a href="index.php?section=problems&amp;id=58" style="text-decoration:none;" title="Click to view problem">
Investigate the number of primes that lie on the diagonals of the spiral grid.
</a></div></td><td><div style="text-align:center;">7388</div></td></tr>

<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>59</b></div></td><td><div><a href="index.php?section=problems&amp;id=59" style="text-decoration:none;" title="Click to view problem">
Using a brute force attack, can you decrypt the cipher using XOR encryption?
</a></div></td><td><div style="text-align:center;">8496</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>60</b></div></td><td><div><a href="index.php?section=problems&amp;id=60" style="text-decoration:none;" title="Click to view problem">
Find a set of five primes for which any two primes concatenate to produce another prime.
</a></div></td><td><div style="text-align:center;">4208</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>61</b></div></td><td><div><a href="index.php?section=problems&amp;id=61" style="text-decoration:none;" title="Click to view problem">
Find the sum of the only set of six 4-digit figurate numbers with a cyclic property.
</a></div></td><td><div style="text-align:center;">4408</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>62</b></div></td><td><div><a href="index.php?section=problems&amp;id=62" style="text-decoration:none;" title="Click to view problem">
Find the smallest cube for which exactly five permutations of its digits are cube.
</a></div></td><td><div style="text-align:center;">5597</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>63</b></div></td><td><div><a href="index.php?section=problems&amp;id=63" style="text-decoration:none;" title="Click to view problem">
How many <i>n</i>-digit positive integers exist which are also an <i>n</i><sup>th</sup> power?
</a></div></td><td><div style="text-align:center;">8840</div></td></tr>

<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>64</b></div></td><td><div><a href="index.php?section=problems&amp;id=64" style="text-decoration:none;" title="Click to view problem">
How many continued fractions for <var>N</var> &#8804; 10000 have an odd period?
</a></div></td><td><div style="text-align:center;">3861</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>65</b></div></td><td><div><a href="index.php?section=problems&amp;id=65" style="text-decoration:none;" title="Click to view problem">
Find the sum of digits in the numerator of the 100<sup>th</sup> convergent of the continued fraction for <i>e</i>.
</a></div></td><td><div style="text-align:center;">5162</div></td></tr>

<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>66</b></div></td><td><div><a href="index.php?section=problems&amp;id=66" style="text-decoration:none;" title="Click to view problem">
Investigate the Diophantine equation <i>x</i><sup>2</sup> &#8722; D<i>y</i><sup>2</sup> = 1.
</a></div></td><td><div style="text-align:center;">3531</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>67</b></div></td><td><div><a href="index.php?section=problems&amp;id=67" style="text-decoration:none;" title="Click to view problem">
Using an efficient algorithm find the maximal sum in the triangle?
</a></div></td><td><div style="text-align:center;">19009</div></td></tr>

<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>68</b></div></td><td><div><a href="index.php?section=problems&amp;id=68" style="text-decoration:none;" title="Click to view problem">
What is the maximum 16-digit string for a "magic" 5-gon ring?
</a></div></td><td><div style="text-align:center;">3625</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>69</b></div></td><td><div><a href="index.php?section=problems&amp;id=69" style="text-decoration:none;" title="Click to view problem">
Find the value of <i>n</i> &le; 1,000,000 for which <i>n</i>/&phi;(<i>n</i>) is a maximum.
</a></div></td><td><div style="text-align:center;">7016</div></td></tr>

<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>70</b></div></td><td><div><a href="index.php?section=problems&amp;id=70" style="text-decoration:none;" title="Click to view problem">
Investigate values of <var>n</var> for which &#966;(<var>n</var>) is a permutation of <var>n</var>.
</a></div></td><td><div style="text-align:center;">4336</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>71</b></div></td><td><div><a href="index.php?section=problems&amp;id=71" style="text-decoration:none;" title="Click to view problem">
Listing reduced proper fractions in ascending order of size.
</a></div></td><td><div style="text-align:center;">6041</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>72</b></div></td><td><div><a href="index.php?section=problems&amp;id=72" style="text-decoration:none;" title="Click to view problem">
How many elements would be contained in the set of reduced proper fractions for <i>d</i> &le; 1,000,000?
</a></div></td><td><div style="text-align:center;">4347</div></td></tr>

<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>73</b></div></td><td><div><a href="index.php?section=problems&amp;id=73" style="text-decoration:none;" title="Click to view problem">
How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?
</a></div></td><td><div style="text-align:center;">5526</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>74</b></div></td><td><div><a href="index.php?section=problems&amp;id=74" style="text-decoration:none;" title="Click to view problem">
Determine the number of factorial chains that contain exactly sixty non-repeating terms.
</a></div></td><td><div style="text-align:center;">5324</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>75</b></div></td><td><div><a href="index.php?section=problems&amp;id=75" style="text-decoration:none;" title="Click to view problem">
Find the number of different lengths of wire that can form a right angle triangle in only one way.
</a></div></td><td><div style="text-align:center;">3119</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>76</b></div></td><td><div><a href="index.php?section=problems&amp;id=76" style="text-decoration:none;" title="Click to view problem">
How many different ways can one hundred be written as a sum of at least two positive integers?
</a></div></td><td><div style="text-align:center;">5944</div></td></tr>

<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>77</b></div></td><td><div><a href="index.php?section=problems&amp;id=77" style="text-decoration:none;" title="Click to view problem">
What is the first value which can be written as the sum of primes in over five thousand different ways?
</a></div></td><td><div style="text-align:center;">3330</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>78</b></div></td><td><div><a href="index.php?section=problems&amp;id=78" style="text-decoration:none;" title="Click to view problem">
Investigating the number of ways in which coins can be separated into piles.
</a></div></td><td><div style="text-align:center;">2949</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>79</b></div></td><td><div><a href="index.php?section=problems&amp;id=79" style="text-decoration:none;" title="Click to view problem">
By analysing a user's login attempts, can you determine the secret numeric passcode?
</a></div></td><td><div style="text-align:center;">11177</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>80</b></div></td><td><div><a href="index.php?section=problems&amp;id=80" style="text-decoration:none;" title="Click to view problem">
Calculating the digital sum of the decimal digits of irrational square roots.
</a></div></td><td><div style="text-align:center;">3888</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>81</b></div></td><td><div><a href="index.php?section=problems&amp;id=81" style="text-decoration:none;" title="Click to view problem">
Find the minimal path sum from the top left to the bottom right by moving right and down.
</a></div></td><td><div style="text-align:center;">6487</div></td></tr>

<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>82</b></div></td><td><div><a href="index.php?section=problems&amp;id=82" style="text-decoration:none;" title="Click to view problem">
Find the minimal path sum from the left column to the right column.
</a></div></td><td><div style="text-align:center;">4088</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>83</b></div></td><td><div><a href="index.php?section=problems&amp;id=83" style="text-decoration:none;" title="Click to view problem">
Find the minimal path sum from the top left to the bottom right by moving left, right, up, and down.
</a></div></td><td><div style="text-align:center;">3327</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>84</b></div></td><td><div><a href="index.php?section=problems&amp;id=84" style="text-decoration:none;" title="Click to view problem">
In the game, Monopoly, find the three most popular squares when using two 4-sided dice.
</a></div></td><td><div style="text-align:center;">2197</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>85</b></div></td><td><div><a href="index.php?section=problems&amp;id=85" style="text-decoration:none;" title="Click to view problem">
Investigating the number of rectangles in a rectangular grid.
</a></div></td><td><div style="text-align:center;">5212</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>86</b></div></td><td><div><a href="index.php?section=problems&amp;id=86" style="text-decoration:none;" title="Click to view problem">
Exploring the shortest path from one corner of a cuboid to another.
</a></div></td><td><div style="text-align:center;">2192</div></td></tr>

<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>87</b></div></td><td><div><a href="index.php?section=problems&amp;id=87" style="text-decoration:none;" title="Click to view problem">
Investigating numbers that can be expressed as the sum of a prime square, cube, and fourth power?
</a></div></td><td><div style="text-align:center;">4144</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>88</b></div></td><td><div><a href="index.php?section=problems&amp;id=88" style="text-decoration:none;" title="Click to view problem">
Exploring minimal product-sum numbers for sets of different sizes.
</a></div></td><td><div style="text-align:center;">1520</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>89</b></div></td><td><div><a href="index.php?section=problems&amp;id=89" style="text-decoration:none;" title="Click to view problem">
Develop a method to express Roman numerals in minimal form.
</a></div></td><td><div style="text-align:center;">3588</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>90</b></div></td><td><div><a href="index.php?section=problems&amp;id=90" style="text-decoration:none;" title="Click to view problem">
An unexpected way of using two cubes to make a square.
</a></div></td><td><div style="text-align:center;">1846</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>91</b></div></td><td><div><a href="index.php?section=problems&amp;id=91" style="text-decoration:none;" title="Click to view problem">
Find the number of right angle triangles in the quadrant.
</a></div></td><td><div style="text-align:center;">3009</div></td></tr>

<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>92</b></div></td><td><div><a href="index.php?section=problems&amp;id=92" style="text-decoration:none;" title="Click to view problem">
Investigating a square digits number chain with a surprising property.
</a></div></td><td><div style="text-align:center;">7959</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>93</b></div></td><td><div><a href="index.php?section=problems&amp;id=93" style="text-decoration:none;" title="Click to view problem">
Using four distinct digits and the rules of arithmetic, find the longest sequence of target numbers.
</a></div></td><td><div style="text-align:center;">2031</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>94</b></div></td><td><div><a href="index.php?section=problems&amp;id=94" style="text-decoration:none;" title="Click to view problem">
Investigating almost equilateral triangles with integral sides and area.
</a></div></td><td><div style="text-align:center;">2151</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>95</b></div></td><td><div><a href="index.php?section=problems&amp;id=95" style="text-decoration:none;" title="Click to view problem">
Find the smallest member of the longest amicable chain with no element exceeding one million.
</a></div></td><td><div style="text-align:center;">2574</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>96</b></div></td><td><div><a href="index.php?section=problems&amp;id=96" style="text-decoration:none;" title="Click to view problem">
Devise an algorithm for solving Su Doku puzzles.
</a></div></td><td><div style="text-align:center;">3430</div></td></tr>

<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>97</b></div></td><td><div><a href="index.php?section=problems&amp;id=97" style="text-decoration:none;" title="Click to view problem">
Find the last ten digits of the non-Mersenne prime: 28433 &times; 2<sup>7830457</sup> + 1.
</a></div></td><td><div style="text-align:center;">12108</div></td></tr>
<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>98</b></div></td><td><div><a href="index.php?section=problems&amp;id=98" style="text-decoration:none;" title="Click to view problem">
Investigating words, and their anagrams, which can represent square numbers.
</a></div></td><td><div style="text-align:center;">1873</div></td></tr>
<tr class="bg_table_row1"><td style="height:40px;"><div style="text-align:center;"><b>99</b></div></td><td><div><a href="index.php?section=problems&amp;id=99" style="text-decoration:none;" title="Click to view problem">
Which base/exponent pair in the file has the greatest numerical value?
</a></div></td><td><div style="text-align:center;">6700</div></td></tr>

<tr class="bg_table_row2"><td style="height:40px;"><div style="text-align:center;"><b>100</b></div></td><td><div><a href="index.php?section=problems&amp;id=100" style="text-decoration:none;" title="Click to view problem">
Finding the number of blue discs for which there is 50% chance of taking two blue.
</a></div></td><td><div style="text-align:center;">2699</div></td></tr>

